Theorem hbxfrbi 1486

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
hbxfrbi.1	|- ( ph <-> ps )
hbxfrbi.2	|- ( ps -> A. x ps )

Assertion

Ref	Expression
hbxfrbi	|- ( ph -> A. x ph )

Proof of Theorem hbxfrbi

Step	Hyp	Ref	Expression
1		hbxfrbi.2	. 2 |- ( ps -> A. x ps )
2		hbxfrbi.1	. 2 |- ( ph <-> ps )
3	2	albii 1484	. 2 |- ( A. x ph <-> A. x ps )
4	1, 2, 3	3imtr4i 201	1 |- ( ph -> A. x ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  hbbi  1562  hb3or  1563  hb3an  1564  hbs1f  1795  hbs1  1957  hbsbv  1960  hbeu1  2055  sb8euh  2068  hbmo1  2083  hbmo  2084  hbab1  2185  hbab  2187  cleqh  2296  hbxfreq  2303  hbral  2526  hbra1  2527